Calculus
From Exampleproblems
I recommend this book: A Course of Modern Analysis by Whittaker and Watson. You may also find this book at Google Books. This book is a hundred years old and is considered the classic calculus book.
Contents 
Derivatives
Definition of Derivative
, provided the limit exists.
For the following problems, find the derivative using the definition of the derivative.
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Power Rule
For the following problems, compute the derivative of with respect to
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solution Derive the power rule for positive integer powers from the definition of the derivative (Hint: Use the Binomial Expansion)
Product Rule
For the following problems, compute the derivative of with respect to
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solution Derive the Product Rule using the definition of the derivative
Quotient Rule
For the following problems, compute the derivative of with respect to x
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solution Derive the Quotient Rule formula. (Hint: Use the Product Rule).
Generalized Power Rule
For the following problems, compute the derivative of with respect to x
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Chain Rule
For the following problems, compute the derivative of with respect to x
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Implicit Differentiation
For the following problems, compute the derivative of with respect to x
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solution where is a function of
Logarithmic Differentiation
For the following problems, compute the derivative of with respect to x
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solution for any functions and where
Second Fundamental Theorem of Calculus
For the following problems, compute the derivative of with respect to x
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solution Give a proof of the theorem
Applications of Derivatives
Slope of the Tangent Line
solution Find the slope of the tangent line to the graph f(x) = 6x when x = 3.
solution Find the slope of the tangent line to the graph f(x) = sin^{2}x when x = π.
solution Find the slope of the tangent line to the graph f(x) = cosx + x^{5} when .
solution Find the equation of the tangent line to the graph f(x) = xe^{x} + x + 5 when x = 0.
solution Find the slope of the tangent line to the graph f(x) = x^{2} when x = 0,1,2,3,4,5,6.
solution Find the slope of the tangent line to the graph x^{2} + y^{2} = 9 at the point (0, − 3).
solution Find the equation of the tangent line to the graph xy = y^{2}x^{2} + y + x + 2 at the point (0, − 2).
Extrema (Maxima and Minima)
solution Find the absolute minimum and maximum on [ − 1,5] of the function f(x) = (1 − x)e^{x}.
solution Find the absolute minimum and maximum on of the function f(x) = sin(x^{2}).
solution Find all local minima and maxima of the function f(x) = x^{2}.
solution Find all local minima and maxima of the function .
solution If a farmer wants to put up a fence along a river, so that only 3 sides need to be fenced in, what is the largest area he can fence with 100 feet of fence?
solution If a fence is to be made with three pens, the three connected sidebyside, find the dimensions which give the largest total area if 200 feet of fence are to be used.
solution Find the local minima and maxima of the function .
Related Rates
solution A clock face has a 12 inch diameter, a 5.5inch second hand, a 5 inch minute hand and a 3 inch hour hand. When it is exactly 3:30, calculate the rate at which the distance between the tip of any one of these hands and the 9 o'clock position is changing.
solution A spherical container of r meters is being filled with a liquid at a rate of . At what rate is the height of the liquid in the container changing with respect to time?
Projectile Motion
Integrals
Riemann Sums
Integration by Substitution
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Integration by Parts
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Trigonometric Integrals
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Trigonometric Substitution
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Partial Fractions
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Special Functions
solution A ball is thrown up into the air from the ground. How high will it go?
solution Let be a continuous function for .
Show that
solution Evaluate
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Applications of Integration
Area Under the Curve
solution Find the area under the curve f(x) = x^{2} on the interval [ − 1,1].
solution Find the total area between the curve f(x) = cosx and the xaxis on the interval [0,2π].
solution Find the area under the curve f(x) = x^{3} + 4x^{2} − 7x + 8 on the interval [0,1].
solution Using calculus, find the formula for the area of a rectangle.
solution Derive the formula for the area of a circle with arbitrary radius r.
Volume
Disc Method
The disc method is a special case of the method of crosssectional areas to find volumes, using a circle as the crosssection.
To find the volume of a solid of revolution, using the disc method, use one of the two formulas below. R(x) is the radius of the crosssectional circle at any point.
If you have a horizontal axis of revolution
If you have a vertical axis of revolution
solution Find the volume of the solid generated by revolving the line y = x around the xaxis, where .
solution Find the volume of the solid generated by revolving the region bounded by y = x^{2} and y = 4x − x^{2} around the xaxis.
solution Find the volume of the solid generated by revolving the region bounded by y = x^{2} and y = x^{3} around the xaxis.
solution Find the volume of the solid generated by revolving the region bounded by y = x^{2} and y = x^{3} around the yaxis.
solution Find the volume of the solid generated by revolving the region bounded by , the xaxis and the line x = 4 around the xaxis.
Shell Method
Crosssectional Areas
solution Find the volume, on the interval , of a 3D object whose crosssection at any given point is a square with side length x^{2} − 9x.
solution Find the volume, on the interval 0 < x < 2π, of a 3D object whose crosssection at any given point is an equilateral triangle with side length .
solution Find the volume of a cylinder with radius 3 and height 10.
Arc Length
solution Calculate the arc length of the curve y = x^{2} from x = 0 to x = 4.
solution Determine the arc length of the curve given by x = t cos t , y = t sin t from t=0
solution Calculate the arc length of y=cosh x from x=0 to x
Mean Value Theorem
solution Find the average value of the function f(x) = e^{2x} on the interval [0,4].
solution Find the average value of the function 2sec^{2}x on the interval .
solution Find the average speed of a car, starting at time 0, if it drives for 5 hours and its speed at time t (in hours) is given by s(t) = 5t^{2} + 7t + e^{t}.
solution Find the average value of the function sin^{6}tcos^{3}t on the interval .
solution Deduce the Mean Value Theorem from Rolle's Theorem.
Series of Real Numbers
Sequences
nth Term Test
If the series converges, then .
Note: This leads to a test for divergence for those series whose terms do not go to 0 but it does not tell us if any series converges.
solution Discuss the convergence or divergence of the series with terms { − 1,1, − 1,1, − 1,1,...}.
solution Discuss the convergence or divergence of the series and .
solution Discuss the convergence or divergence of the series .
Telescopic Series
If {a_{k}} is a convergent real sequence, then .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
Geometric Series
The series converges if − 1 < r < 1 and, moreover, it converges to . For any other value of r, the series diverges. More generally, the finite series, for any value of r.
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
Integral Test
If the function f is positive, continuous, and decreasing for , then
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converge together or diverge together.
Notice that if f were negative, continuous, and increasing this is also true since such a function would simply be the negative of some function which is positive, continuous, and decreasing and multiplying by 1 will not change the convergence of a series.
solution Discuss the convergence of .
solution Discuss the convergence of .
solution Discuss the convergence of .
solution Discuss the convergence of .
solution Explain why the integral test is or is not applicable to .
solution Explain why the integral test is or is not applicable to .
solution Explain why the integral test is or is not applicable to .
solution Discuss the convergence of .
solution Discuss the convergence of .
solution Discuss the convergence of .
Comparison of Series
 

solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
Dirichlet's Test
Let for .
If the sequence of partial sums is bounded and as , then converges.
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series where a_{k} = {7,4,6,3, − 10, − 10,7,4,6,3, − 10, − 10,...}.
solution Discuss the convergence or divergence of the series for any integer .
solution Discuss the convergence or divergence of the series where {a_{k}} = {1, − 1,1,2, − 3,1,2,4, − 7,1, − 1,1,2, − 3,1,2,4, − 7,...}.
Alternating Series
If a_{n}, then the alternating series
and
converge if the absolute value of the terms decreases and goes to 0.
solution Discuss the convergence of the series .
solution Discuss the convergence of the series .
solution Discuss the convergence of the series .
solution Discuss the convergence of the series .
solution Discuss the convergence of the series .
solution Discuss the convergence of including whether the sum converges absolutely or conditionally.
solution Discuss the convergence of including whether the sum converges absolutely or conditionally.
solution Discuss the convergence of including whether the sum converges absolutely or conditionally.
Ratio Test
For the infinite series ,
1. If , then the series converges absolutely.
2. If , then the series diverges.
3. If , then the test is inconclusive.
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
solution Discuss the convergence or divergence of the series .
Root Test
Let and
1. If , then converges absolutely.
2. If , then diverges.
3. If , this test is inconclusive.
solutionDiscuss the convergence or divergence of the series .
solutionDiscuss the convergence or divergence of the series .
solutionDiscuss the convergence or divergence of the series .
solutionDiscuss the convergence or divergence of the series .
solutionDiscuss the convergence or divergence of the series .
solutionDiscuss the convergence or divergence of the series .
solutionDiscuss the convergence or divergence of the series .
solutionDiscuss the convergence or divergence of the series .
Cauchy Condensation Test
The series and converge or diverge together.
solutionDiscuss the convergence or divergence of the series .
solutionDiscuss the convergence or divergence of the series .
solutionDiscuss the convergence or divergence of the series .
solutionDiscuss the convergence or divergence of the series .
solutionDiscuss the convergence or divergence of the series , where n is some real number.
Logarithmic Test
Raabe's Test
Series of Real Functions
solution Find the infinite series expansion of
solution Investigate the convergence of this series:
solution Investigate the convergence of this series:
solution Find the upper limit of the sequence
solution Find the upper limit of the sequence
solution Evaluate
solution Evaluate .
solution Find the upper limit of the sequence
solution Find the upper limit of the sequence
solution Evaluate
solution Determine the interval of convergence for the power series